On a generalized notion of metrics

In these notes we generalize the notion of a (pseudo) metric measuring the distance of two points, to a (pseudo) n-metric which assigns a value to a tuple of n points. We present two principles of constructing pseudo n-metrics. The first one uses the Vandermonde determinant while the second one uses exterior products and is related to the volume of the simplex spanned by the given points. We show that the second class of examples induces pseudo n-metrics on the unit sphere of a Hilbert space and on matrix manifolds such as the Stiefel and the Grassmann manifold. Further, we construct a pseudo n-metric on hypergraphs and discuss the problem of generalizing the Hausdorff metric for closed sets to a pseudo n-metric.


Introduction
The classical notion of a (pseudo) metric or distance on a set X is a map which assigns to any two points in X a real value such that d is (semi-)definite, symmetric, and satisfies the triangle inequality.Since the subject is vast we just mention [11] as an elementary and [6] as an advanced monograph.The purpose of this contribution is to investigate an n-dimensional (n 2) generalization of this notion.To be precise, we call a map a pseudo n-metric on X if it has the following properties: (i) (Semidefiniteness) If x = (x 1 , . . ., x n ) ∈ X n satisfies x i = x j for some i = j, then d(x 1 , . . ., x n ) = 0. (ii) (Symmetry) For all x = (x 1 , . . ., x n ) ∈ X n and all π ∈ P n d(x π(1) , . . ., x π(n) ) = d(x 1 , . . ., x n ). (1.1) Here P n denotes the group of permutations of {1, . . ., n}. (iii) (Simplicial inequality) For all x = (x 1 , . . ., x n ) ∈ X n and y ∈ X d(x 1 , . . ., x n ) n i=1 d(x 1 , . . ., x i−1 , y, x i+1 , . . ., x n ). (1.2) Clearly, a pseudo 2-metric agrees with the notion of an ordinary pseudo metric.If we sharpen condition (i) to definiteness, i.e. d(x 1 , . . ., x n ) = 0 implies x i = x j for some i = j, then we call d an n-metric.
Due to its various areas of occurrence, we think that the axiomatic notion of a pseudo n-metric deserves to be studied in its own right.

Basic properties and the Vandermonde example
In this secton we collect some basic constructions and properties of pseudo nmetrics.Our main example is based on the Vandermonde determinant which leads to a pseudo n-metric that is homogeneous of degree 1  2 n(n − 1).In the following, a mapping d : X n → R is called nonnegative if d(x) 0 holds for all x ∈ X n .Lemma 2.1.
(i) A pseudo n-metric on X is nonnegative.(ii) If the mapping d : X n → R is nonnegative, semidefinite, symmetric, and satisfies (1. 2) for all pairwise different elements y, x 1 , . . ., x n , then d is a pseudo n-metric.
Proposition 2.2.For a pseudo n-metric d on a set X the following holds: (i) d X defines a pseudo n-metric on any subset Y ⊂ X.
(ii) If d Y is a pseudo n-metric an a set Y , then defines a pseudo n-metric on X × Y for any monotone norm • in R 2 .(iii) Let F ⊆ X Y be a subset of functions from some set Y to X.Further assume that a normed space Z ⊆ R Y is given with a monotone norm • Z and the following property: Then a pseudo n-metric on F is defined by Proof.Since most assertions are obvious we consider only the simplicial inequality in cases (ii) and (iii).Recall that a norm • Z on a linear space Z of real-valued functions f : Y → R is called monotone (see e.g.[10,Ch.5.4]) iff for all In case (ii) one takes norms in the following vector inequality for ξ ∈ X, η ∈ Y d X (x 1 , . . ., x n ) d Y (y 1 , . . ., y n ) The proof of (iii) is analogous.For f 1 , . . ., f n , g ∈ F observe the pointwise inequality which by the monotonicity of the norm implies Our basic example of this section employs the well-known Vandermonde determinant associated with numbers z j ∈ C, j = 1, . . ., n (see [10, Ch.0.9]): defines an n-metric in C and by restriction also in R.
Proof.According to Proposition 2.2(i) it suffices to consider (2.3) in C. By the definition (2.3) it is clear that d V is nonegative, semidefinite, and even definite.The symmmetry follows from By Lemma 2.1(ii) it is enough to show the simplicial inequality for pairwise different numbers z 1 , . . ., z n , y.Then there is a unique solution (a 1 , . . ., a n ) ⊤ ∈ C n to the linear system    The solution is given by Cramer's rule as follows with the Vandermonde determinant from (2.2).The proof is completed by the following estimate (2.4) Remark 2.4.It is remarkable that the proof yields the following more general inequality Let us also note that a straightforward generalization of (2.3) to a product of norms fails in higher dimensional spaces; see Section 6.1.However, in Section 6.1 we present an alternative generalization of the Vandermonde pseudo n-metrics.
In case n = 3 we investigate when the simplicial inequality holds with equality.
Proposition 2.5.For the 3-metric holds with pairwise different numbers y, z 1 , z 2 , z 3 ∈ C if and only if the quadruple (y, z 1 , z 2 , z 3 ) belongs (up to a shift and a multiplication by a complex number and up to a permutation of z 1 , z 2 , z 3 ) to the following two parameter family (q, s > 0): Further, the equality (2.5) holds with Proof.From (2.4) we find that equality holds in (1.2) if and only if Since the numbers are distinct we may shift y to zero and multiply by a complex number such that z 1 = 1.Recall that the equality holds if and only if there exists an index j ∈ {1, . . ., n} and real numbers c i 0 such that z i = c i z j for i = 1, . . ., n.Moreover, if z i = 0 for all i = 1, . . ., n then the latter property holds for any j ∈ {1, . . ., n} with numbers c i > 0. We apply this to (2.7) with n = 3 after normalizing y = 0 and z 1 = 1.Thus the equality (2.5) holds if and only if there are numbers c 2 , c 3 > 0 such that Since z 2 , z 3 = 0 this is equivalent to the system Subtract the second from the first equation and use z 2 = z 3 to find the equivalent system 1 With the last equation we eliminate z 3 = −c −1 3 (1 + c 2 z 2 ) from the first to obtain the quadratic equation The solutions to (2.8) are then given by Introducing the parameters s = c 2 + c 3 > 0 and q = c3 c2 > 0 we may write this as Finally, we observe the relation where the latter is a permutation of (z + 2 , z − 3 )(q, s).Hence the family (2.6) covers all solutions up to permutation.
For the normalized coordinates y = 0, z 1 = 1 the condition 1 = |z 2 | = |z 3 | holds if and only if s = 2, q = 1.For these values the numbers ) are the third roots of unity which belong to the equilateral case.Equality holds if and only if the triangle is equilateral.Neither did we find a reference to this fact in the literature nor an elementary geometric proof.
Combining Proposition 2.2 and Theorem 2.3 leads to the following examples of pseudo n-metrics.
Example 2.8.Let (Ω, G, µ) be a bounded measure space.We apply Proposition 2.2 to Z = L p (Ω, G; R) with the monotone norm • L p and to F = L r (Ω, G; R) with 2r n(n − 1)p.Then the expression . defines a pseudo n-metric on L r (Ω, G; R).Note that 2r n(n − 1)p guarantees that the property (2.1) holds.

Examples in linear spaces
3.1.Pseudo n-norms.In vector spaces it is natural to first define a pseudo nnorm and then define a pseudo n-metric by taking the n-norm of differences.
(iii) (Positive homogeneity) For all x = (x 1 , . . ., x n ) ∈ X n and for all λ ∈ R (iv) (Multi-sublinearity) For all x = (x 1 , . . ., x n ) ∈ X n and y ∈ X, Clearly, a pseudo 1-norm is a semi-norm in the usual sense.Also note that the positive homogeneity (3.2) and the sublinearity (3.3) transfers from the first component to all other components via (3.1).
Next consider a matrix A ∈ R n,n and the induced linear map Remark 3.3.The result shows that the pseudo n-norm is a volume form on X n .
Proof.If A is singular then the vectors (A X (x 1 , . . ., x n )) j , j = 1, . . ., n are linearly dependent, hence formula (3.5) is trivially satisfied.Otherwise there exists a decomposition A = P LDU where P is an n × n permutation matrix, L ∈ R n,n resp.U is a lower resp.upper triangular n × n-matrix with ones on the diagonal and show that all 4 types of matrices satisfy the formula (3.5).For P X this follows from (3.1) and for With (3.3) and Definition 3.1(i) we obtain (⋆, By induction one eliminates all terms L i,j x i , i > j from the pseudo n-norm.Since det(L) = 1 this proves (3.5) for L. In a similar manner, one shows and finds that (3.5) is also satisfied for the map R X .
Proposition 3.4.Let • be a pseudo (n − 1)-norm on a vector space X.Then defines a pseudo n-metric on X.
Proof.Let x i = x j for some i, j ∈ {1, . . ., n}, i = j.Then the vectors x ν − x 1 , ν = 2, . . ., n are linearly dependent, hence d(x 1 , . . ., x n ) = 0 follows from condition (i) of Definition 3.1.Since P n is generated by transpositions, it is enough to show (1.1) when transposing x 1 with some x j , j ∈ {2, . . ., n}.For this purpose consider the matrix with −1 in the j-th row Taking norms and using (3.1) and (3.5) leads to For the simplicial inequality (1.2) we use the multi-sublinearity (3.3) For the last term we used (3.5) and for the upper triangular matrix Thus we have proved for j = 2 the following assertion In the induction step we estimate the last term further by splitting As in the first step we can replace all vectors x i − x 1 , i j + 2 in the last term by x i − y.This proves that (3.6) holds for j + 1.For j = n equation (3.6) reads X) is the set of continuous alternating k-linear forms on the dual X ⋆ , using the identification of X and its bidual X ⋆⋆ .
k X is called the k-fold exterior product of X. Elements of k (X) are exterior products x 1 ∧ . . .∧ x k defined for x 1 , . . ., x k ∈ X by the dual pairing where f 1 , . . ., f k ∈ X ⋆ and •, • is the dual pairing of X ⋆ and X.As usual we write in the following k i=1 By linearity one can extend exterior products to sums Closing the linear hull with respect to the norm turns k (X) into a Banach space.The following lemma is elementary (see [2, Lemma 3.2.6])Lemma 3.5.Let k X be the k-fold exterior product of a separable reflexive Banach space space X with k dim(X).Then the following holds (i) If X is m-dimensional (m k) and e 1 , . . ., e m is a basis of X then is a basis of k X.In particular k X has dimension m k .(ii) One has x 1 ∧ . . .∧ x k = 0 if and only if x 1 , . . ., x k are linearly dependent.(iii) If X is a Hilbert space with inner product •, • then the bilinear and continuous extension of defines an inner product on the Hilbert space k X .In particular, the corresponding norm is the volume of the k-dimensional parallelepiped spanned by x 1 , . . ., x k .Further, the generalized Hadamard inequality holds for j = 1, . . ., k (3.9) Proposition 3.6.
(i) Let n X be the n-fold exterior product of a separable reflexive Banach space X and let • be a norm in n X.Then defines an n-norm on X. (ii) Let n−1 X be the (n − 1)-fold exterior product of a Banach space X and let • be a norm in n−1 X.Then defines a pseudo n-metric on X.One has d(x 1 , . . ., x n ) = 0 if and only if (3.11) The corresponding pseudo 3-metric on H is

Some pseudo n-metrics on manifolds
We use the results from the previous section to set up a pseudo n-metric on the unit sphere of a Hilbert space.Proposition 4.1.Let (H, •, •, H , • H ) be a Hilbert space and consider the unit sphere Then the n-norm defined by (3.10) and (3.8) generates a pseudo n-metric for x i ∈ S H , i = 1, . . ., n as follows Proof.The semidefiniteness and the symmetry are obvious consequences of Proposition 3.6.It remains to prove the simplicial inequality.For given x i ∈ S H , i = 1, . . ., n we write y ∈ S H with suitable coefficients c, c i (i = 1, . . ., n) as Then we have the equality Using | x i , x j H | 1 we obtain From (4.1) and the properties of the exterior product we have for i = 1, . . ., n Further, the orthogonality in (4.1) implies via (3.7) hence by (4.1) where we use the abbreviations . Note that Hadamard's inequality (3.9) implies d n d i,n−1 for i = 1, . . ., n.With (4.2), (4.3) and the triangle inequality in R 2 we find Remark 4.3.The value d(u, v, w) in (4.4) agrees with the three-dimensional polar sine 3 polsin(O, u v w), already defined by Euler, see [8,Sect.6].The work [8] discusses various ways of measuring n-dimensional angles and its history in spherical geometry.The main result is a simple derivation of the law of sines for the ndimensional sine n sin and the n-dimensional polar sine n polsin defined by However, the simplicial inequality in Proposition 4.1 seems not to have been observed.
Next we consider the Hilbert space H = L(R k , R m ) of linear mappings from R k to R m endowed with the Hilbert-Schmidt (or Frobenius) inner product and norm Here the vectors e j , j = 1, . . ., k form an orthonormal basis of R k and the prefactor 1 k is used for convenience, so that an orthogonal map A ∈ H, i.e.A ⋆ A = I k , satisfies A H = 1.It is well known that the Hilbert-Schmidt inner product and norm are independent of the choice of orthonormal basis (e j ) k j=1 .By Proposition 4.1 the unit sphere S H in H carries a pseudo n-metric defined by Example 4.5.In case n = 2 we obtain For the Grassmannian G(k, m) of k-dimensional subspaces of R m , k m, the situation is not so simple.One can identify G(k, m) with the quotient space St(k, m)/ ∼, where A ∼ B ⇐⇒ ∃Q ∈ O(k) : A = BQ (4.9) by setting V = range(A) for [A] ∼ ∈ St(k, m)/ ∼.Then the Hilbert Schmidt norm is invariant w.r.t. the equivalence class, but the inner product (4.6) is not, since we obtain terms AQ 1 e j , BQ 2 e j 2 for which the orthogonal maps Q 1 , Q 2 differ, in general.Therefore, we associate with every element V ∈ G(k, m) the orthogonal projection P onto V , given by P = AA ⋆ where V = range(A) for A ∈ St(k, m).The projection is an invariant for the equivalence classes in (4.9).It is appropriate to measure orthogonal projections of rank k by the scaled Hilbert-Schmidt inner product and norm where (e j ) j=1,...,m form an orthonormal basis of R m .We claim that the orthogonal projection P belongs to the unit ball To see this, choose a special orthonormal basis of R m where e 1 , . . ., e k are the columns of A and e k+1 , . . ., e m form a basis of the orthogonal complement V ⊥ .Since the norm is invariant and AA * e j = e j for j = 1, . . ., k, AA ⋆ e j = 0 for j > k we obtain P k,H = 1.Thus we can proceed as before and invoke Propositions 4.1 and 2.2(i) to obtain the following result.
. ., n be the corresponding orthogonal projections.Then the setting defines a pseudo n-metric on G(k, m).
For an interpretation of (4.10) let us compute the inner product of two orthogonal projections.
For n = 2 this leads to an explicit expression of (4.10). where are the principal angles of the subspaces V 1 and V 2 .Proof.From Lemma 4.7 and (4.10) we find In Section 6.2 we continue the discussion of pseudo n-metrics on the Grassmannian and its relation to known 2-metrics.

A pseudo n-metric on hypergraphs
The notion of hypergraph allows edges which connect more than two vertices; see [5].
Definition 5.1.A pair (V, E) is called a hypergraph if V is a finite set and E is a subset of the power set P(V ) of V .An element e ∈ E is called a hyperedge.In particular, it is called an n-hyperedge if #e = n.The hypergraph is called n-uniform if all its hyperedges are n-hyperedges.
Obviously, a 2-uniform hypergraph is an ordinary (undirected) graph.The following definition generalizes the standard notion of connectedness.Definition 5.2.Let (V, E) be an n-uniform hypergraph.
(i) A subset P ⊂ E is called a connected component of (V, E) if for any two e 0 , e ∞ ∈ P there exist finitely many e 1 , . . ., e k ∈ P such that e i−1 ∩ e i = ∅ for i = 1, . . ., k + 1 where e k+1 := e ∞ .Any subset W ⊂ e∈P e is said to be connected by P .
and V = e∈E e.
In case n = 2 this agrees with the usual notion of connected components.For a connected hypergraph one can connect any subset of V by the (maximal) connected component E. Remark 5.6.In case n = 2 this generates the topology on hypergraphs discussed in [7].
In the following we pursue another highdimensional generalization of the Vandermonde expression (2.3) which uses multilinear symmetric maps.It is based on the following purely algebraic result.Theorem 6.2.(The general Vandermonde equalities) Let X, Y be vector spaces over K = R, C and let be an M n -linear and symmetric map.
(i) For all x 1 , . . ., x n ∈ X the following holds: (ii) The quantity satisfies for all ξ ∈ X and x 1 , . . ., x n ∈ X the following equality: Proof.First note that it is convenient to write the arguments of the symmetric map A as products since the sequence of arguments does not matter. (i): For n = 2 the equation (6.2) is obvious: ) + sign(2, 1)A(x For the induction step from n to n + 1 we use the symmetry of A: We consider the summand for any two indices ℓ, k ∈ {1, . . ., n} with π(ℓ) = π(k)+ 1 (hence ℓ = k) and any σ ⊆ {1, . . ., n} with k ∈ σ, ℓ / ∈ σ.Let τ k,ℓ be the transposition of k and ℓ and set .
Proof.The semidefiniteness and the nonnegativity are obvious from the definition (6.6).The prove symmetry we apply the representation (6.2) for every σ ∈ P n : Finally, the simplicial inequality follows by taking norms in (6.3) and using the triangle inequality.
The n-mterics from Theorem 2.3 are special cases of Corollary 6.3 for dim(X) = dim(Y ) = 1, 2 when taking the multilinear form A induced by real resp.complex multiplication.It will be of interest to see whether true n-metrics can be constructed via Corollary 6.3 for spaces with dim(X) 3.

6.2.
More pseudo n-metrics on the Grassmannian G(k, m).The pseudo nmetric (4.10) on G(k, m) is not completely satisfactory for several reasons.First, one expects a suitable pseudo n-metric on G(k, m) to be a canonical generalization of the standard 2-metric given by where P 1 , P 2 denote the orthogonal projections onto V 1 , V 2 respectively, and • 2 denotes the Euclidean operator norm (spectral norm) in R m ; see [9,Ch.6.4.3].This 2-metric may be expressed equivalently as where θ k is the largest principal angle between V 1 and V 2 ; see [9, Ch.2.5, Ch.6.4] and Proposition 4.8.Note that the 2-metric (6.7) differs from the 2-metric in (4.11).An alternative to (4.10) is to measure the exterior product n j=1 P j of orthogonal projections by a norm different from the Hilbert-Schmidt norm (cf.(4.5)): , where • * is either the spectral norm • 2 or its dual • D given by the sum of singular values; see [10,Ch.5.6].However, neither were we able to prove the simplicial inequality for this setting, nor did we find an explicit expression for the associated 2-metric comparable to (6.7).
The fact that the Grassmannian may be viewed as a quotient space of St(k, m) (see (4.9)) suggests another construction of a 2-metric on G(k, m) by setting for V j = range(A j ), A j ∈ St(k, m) and d H from (4.7), (4.8).The symmetry and definiteness of d G,H are obvious.The triangle inequality is also easily seen: for Then we obtain The next proposition gives an explicit expression of the 2-metric (6.8) in terms of the principal angles θ j , j = 1, . . ., k between the subspaces V 1 and V 2 .
Note that (6.9) differs from both expressions (6.7) and (4.11).Let us further note that we did not succeed to prove the simplicial inequality for the natural generalization of (6.8) We rather conjecture that this is false which is suggested by analogy to a counterexample from the next subsection.Finally, we discuss the special case k = 1, when we can identify V ∈ G(1, m) with a unit vector v ∈ S R m such that V = span{v}.Proposition 4.1 then shows that a suitable pseudo n-metric on G(1, m) is given by which is consistent with (6.7).On the other hand the pseudo n-metric (4.10) leads to which in case n = 2 differs from (6.7).Therefore, we still consider it an open problem to construct a pseudo n-metric on the Grassmannian which is a natural generalization of the 2-metric (6.7).
6.3.Pseudo n-metric for subsets.It is natural to ask whether the classical Hausdorff distance for closed sets of a metric space (see e.g.[3, Chapter 1.5]) has an extension to a pseudo n-metric.Definition 6.5.Let d : X n → R be a pseudo n-metric on a set X. Then we define for nonempty subsets A j ⊆ X, j = 1, . . ., n the following quantities For an ordinary 2-metric the construction (6.10) leads to the familiar Hausdorff distance It is well known that d H is a metric on the set of all closed subsets of X where 'closedness' is defined with respect to the given metric in X; see e.g.[3,Ch.9.4].
It is not difficult to see that the map d H defined in (6.10) is semidefinite and symmetric .While the simplicial inequality is true for n = 2, i.e. the Hausdorff distance, we claim that it is generally false for n 3. Example 6.6.(Counterexample for the simplicial inequality) Consider the case n = 3 and four finite sets For convenience we introduce the following abbreviations for i = 1 and j, k, ℓ = 1, . . ., N : The values of the 3-metric with three arguments from different sets A ν in X are defined as follows: Our first observation is which implies d ijk0 d ij0ℓ + d i0kℓ + d 0jkℓ =: S ℓ for all i, j, k, ℓ.In fact, we find Further, the simplicial inequalities This contradicts the simplicial inequality (1.2).
In the following we define d for all triples where at least two elements lie in the same set, so that the axioms of a 3-metric are satisfied.Without loss of generality we define d(ξ 1 , ξ 2 , ξ 3 ) when ξ 1 , ξ 2 , ξ 3 lie in sets A 1 , A 2 , A 3 , A 4 with increasing lower index.All other values are then given by the permutation symmetry (1.1).Further, we set d(ξ 1 , ξ 2 , ξ 3 ) = 0 if two of the arguments agree (semidefiniteness) or if all three arguments ξ 1 , ξ 2 , ξ 3 lie in the same set A ν .It remains to define the following quantities For the last line note that d 00,kκ,ℓ = 1 if k = ℓ and d 00,kκ,λ = 1 if k = λ.
For the last line note that d 0,jj ′ ,0ℓ = 1 if j = ℓ and d 0,ιj ′ ,0ℓ = 1 if ι = ℓ.By inspection one finds that the remaining four cases can be handled in the same way.
While this example shows that the definition (6.10) with an arbitrary pseudo n-metric does not necessarily define a pseudo n-metric on subsets, it is still possible that some of the special n-metrics from Section 3 will have this property.

Remark 2 . 6 .
A simple consequence of Theorem 2.3 and Proposition 2.5 for y = 0, |z 1 | = |z 2 | = |z 3 | = 1 is the following elementary statement: The sides a, b, c of a triangle with vertices on the unit circle satisfy the inequality abc a + b + c.

Example 4 . 2 .
As examples we list the induced pseudo 2-metric (compare (3.11)) and the pseudo 3-metric on S H :
3.2.Construction via exterior products.The previous results suggest to use exterior products in defining appropriate pseudo n-norms.For this purpose recall the following calculus for a separable reflexive Banach space X; see [15, Ch.V], [1, Section 6], [2, Ch.3.2.3].The linear space k n are linearly dependent.Proof.By Proposition 3.4 it suffices to show that (3.10) defines an n-norm on X. Condition (i) in Definition 3.1 follows from Lemma 3.5 (ii), and (3.1) is a consequence of the alternating property of the exterior product.Further, the positive homogeneity (3.2) follows from the homogeneity of the exterior product and the positive homogeneity of the norm in n X.Finally, inequality (3.3) is implied by taking norms of the multilinear relation (x 1 + y) ∧